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Continuity of Solutions of a Class of Fractional Equations

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Abstract

In practice many problems related to space/time fractional equations depend on fractional parameters. But these fractional parameters are not known a priori in modelling problems. Hence continuity of the solutions with respect to these parameters is important for modelling purposes. In this paper we will study the continuity of the solutions of a class of equations including the Abel equations of the first and second kind, and time fractional diffusion type equations. We consider continuity with respect to the fractional parameters as well as the initial value.

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References

  1. Bazhlekova, E.: The abstract Cauchy problem for the fractional evolution equation. Frac. Calc. Appl. Anal. 1(3), 255–270 (1998)

    MathSciNet  MATH  Google Scholar 

  2. Bondarenko, A.N., Ivaschenko, D.S.: Numerical methods for solving problems for time fractional diffusion equation with variable coefficient. J. Inverse Ill-Posed Probl. 17, 419–440 (2009)

    MathSciNet  MATH  Google Scholar 

  3. del Castillo Negrete, D., Carreras, B.A., Lynch, V.E.: Fractional diffusion in plasma turbulence. Phys. Plasmas 11(8), 3854–3864 (2004)

    Article  Google Scholar 

  4. del Castillo Negrete, D., Carreras, B.A., Lynch, V.E.: Nondiffusive transport in plasma turbulence: A fractional diffusion approach. Phys. Rev. Lett. 94, 065003 (2005)

    Article  Google Scholar 

  5. Clément, P., Gripenberg, G., Londen, S.-O.: Schauder estimates for equations with fractional derivatives. Trans. Amer. Math. Soc. 352(5), 2239–2260 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Clément, P., Londen, S.-O., Simonett, G.: Quasilinear evolutionary equations and continuous interpolation spaces. J. Diff. Eqs. 196, 418–447 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ren, C., Xu, X., Lu, S.: Regularization by projection for a backward problem of the time-fractional diffusion equation. J. Inverse Ill-Posed Probl. 22(1), 121–139 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chen, Z.-Q., Meerschaert, M.M., Nane, E.: Space-time fractional diffusion on bounded domains. J. Math. Anal. Appl. 393(2), 479–488 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chechkin, A.V., Gorenflo, R., Sokolov, I.M.: Fractional diffusion in inhomogeneous media. J. Phys. A 38(42), L679–L684 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cheng, J., Nakawaga, J., Yamamoto, M., Yamazaki, T.: Uniqueness in an inverse problem for a one-dimensional fractional diffusion equation. Inverse Probl. 25, 115002 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ding, Y., Ye, H.: A fractional-order differential equation model of HIV infection of CD4+ T-cells. Math. Comput. Model 50(34), 386–392 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Djordjevic, I.V.D., Jaric, I.J. , Fabry, B., Fredberg, J.J., Stamenovic, I.D.: Fractional derivatives embody essential features of cell rheological behavior. Ann. Biomed. Eng. (2003)

  13. Engel, K.-J., Nagel, R.: One-parameter Semigroup for Linear Evolution Equations. Springer-Verlag, New York (2000)

    MATH  Google Scholar 

  14. Ginoa, M., Cerbelli, S., Roman, H.E.: Fractional diffusion equation and relaxation in complex viscoelastic materials. Phys. A 191, 449—453 (1992)

    Google Scholar 

  15. Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions, Related Topics and Applications. Springer-Verlag, Berlin (2014)

    Book  MATH  Google Scholar 

  16. Gorenflo, R., Vessella, S.: Abel Integral Equations, Analysis and Applications, Lecture Notes in Mathematics 1461. Springer-Verlag, New York (1991)

    MATH  Google Scholar 

  17. Hatano, Y., Nakagawa, J., Wang, S., Yamamoto, M.: Determination of order in fractional diffusion equation. J. Math-for-Industry 5, 51–57 (2013)

    MathSciNet  MATH  Google Scholar 

  18. Kim, S., Kavvas, M.L.: Generalized Fick’s law and fractional ADE for pollution transport in a river: Detailed derivation. J. Hydrol. Eng. 11(1), 80–83 (2006)

    Article  Google Scholar 

  19. Lions, J.L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969)

    MATH  Google Scholar 

  20. Li, X., Xu, C.: A space-time spectral method for the time fractional diffusion equation. SIAM J. Numer. Anal. 47(3), 2108–2131 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Li, G., Zhang, D., Jia, X., Yamamoto, M.: Simultaneous inversion for the space-dependent diffusion coefficient and the fractional order in the time-fractional diffusion equation, vol. 29 (2013)

  22. Liu, J.J., Yamamoto, M.: A backward problem for the time-fractional diffusion equation. Appl. Anal. 89, 1769–1788 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Meerschaert, M.M., Nane, E., Vellaisamy, P.: Fractional Cauchy problems on bounded domains. Ann. Probab. 37, 979–1007 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  25. Miller, K.S., Samko, S.G.: Completely monotonic functions. Integral Transforms Spec. Funct. 12, 389–402 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. Nigmatulin, R.: The realization of the generalized transfer equation in a medium with fractal geometry. Phys. Stat. Sol. B 133, 425–430 (1986)

    Article  Google Scholar 

  27. Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    MATH  Google Scholar 

  28. Rodrigues, F.A., Orlande, H.R.B., Dulikravich, G.S.: Simultaneous estimation of spatially dependent diffusion coefficient and source term in a nonlinear 1D diffusion problem. Math. Comput. Simul. 66, 409–424 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sabatelli, L., Keating, S., Dudley, J., Richmond, P.: Waiting time distribution in financial markets. Eur. Phys. J. 27, 273–275 (2002)

    MathSciNet  Google Scholar 

  30. Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382(1), 426–447 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Scalas, E., Gorenflo, R., Mainardi, F.: Fractional calculus and continuous-time finance. Phys. A 284, 367–384 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. Simon, T.: Comparing Fréchet and positive stable laws. Electron. J. Probab. 19(16), 1–25 (2014)

    MathSciNet  MATH  Google Scholar 

  33. Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  34. Zacher, R.: Weak solution of abstract evolutionary integro-differential equations in Hilbert spaces. Funkcialaj Ekvacioj 52, 1–18 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zheng, G.H., Wei, T.: Two regularization methods for solving a Riesz-Feller space-fractional backward diffusion problem. Inverse Probl. 26(11), 115017 (2010). 22 pp

    Article  MathSciNet  MATH  Google Scholar 

  36. Zygmund, A.: Trigonometric Series II. Cambridge University Press, Cambrige (1959)

    MATH  Google Scholar 

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Acknowledgments

We would like to thank the anonymous referee for reading the paper carefully and finding many errors in the earlier version of our paper. These comments helped us improve the paper. This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2016.26.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Science, Vietnam National University HCMC, Ho Chi Minh City, Viet Nam

    Duc Trong Dang, Dang Minh Nguyen & Nguyen Huy Tuan

  2. Department of Mathematics and Statistics, Auburn University, Auburn, AL, USA

    Erkan Nane

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  1. Duc Trong Dang
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  2. Erkan Nane
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  3. Dang Minh Nguyen
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  4. Nguyen Huy Tuan
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Correspondence to Erkan Nane.

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Dang, D.T., Nane, E., Nguyen, D.M. et al. Continuity of Solutions of a Class of Fractional Equations. Potential Anal 49, 423–478 (2018). https://doi.org/10.1007/s11118-017-9663-5

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  • DOI: https://doi.org/10.1007/s11118-017-9663-5

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